Respuesta :
Answer : 2446 years.
Explanation :
Length of semi major axis is, [tex]a=84\ au= 1.496\times 10^{11}\ m[/tex]
According to Kepler's third law, square of time period of an orbit is directly proportional to the cube of the semi major axis.
i.e [tex]T^2=\dfrac{4\pi^2}{GM}a^3[/tex]
where G is gravitational constant
M is mass of sun, [tex]M=1.98\times 10^{30}\ Kg[/tex]
So, [tex]T^2=\dfrac{4\times (3.14)^2}{6.6\times 10^{-11}Nm^2/Kg\times 1.98\times 10^{30}\Kg}[/tex]
[tex]T^2=3\times 10^{-19}\times(84\times 1.496\times 10^{11})^3[/tex]
[tex]T^2=3\times 10^{-19}\times 1984415.6\times 10^{33}[/tex]
[tex]T^2=59532469.8\times 10^{14}\ s[/tex]
[tex]T=7715.7\times 10^7\ s[/tex]
since, [tex]1\ sec=3.17\times 10^{-8}\ years[/tex]
So, orbital period is approximately 2446 years.
